- Data are observations or measurements. Categorical data describe groups; numerical data are quantities.
- Frequency counts how often a value or category appears.
- Arithmetic mean is the sum of observations divided by their count.
- Sort the data; take the middle value, or average the two middle values for even-sized datasets.
- The mode is the most frequent value; data may have one, multiple or no unique mode.
- Range equals maximum minus minimum and measures overall spread.
- A weighted mean uses the sum of value×weight divided by total weight.
- Combined mean=(n₁m₁+n₂m₂)/(n₁+n₂), where n values are group sizes.
- The unweighted average of two group means is correct only for equal group sizes.
- Adding c to each observation increases the mean and median by c; multiplying by positive k scales them by k.
- Extreme values can affect the mean strongly while the median is often less sensitive.
- A bar chart displays frequencies of categories; a clear axis and scale prevent misleading comparisons.
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
How Quantities Combine: Understanding Data — Class 9 Mathematics
Textbook edition: Ganita Manjari 2026–27, Part II, Chapter 10. This belongs to the Ganita Manjari (2026–27) sequence.
Introduction: Data are observations or measurements. Categorical data describe groups; numerical data are quantities.
Original supplementary notes: the precise numbered textbook exercises, diagrams and activities have not been verified page-by-page against the complete official NCERT textbook. The following practice questions are original, not represented as verbatim NCERT exercise solutions.
Learning objectives
- Explain concepts and definitions accurately.
- Apply formulas and algorithms with stated conditions.
- Interpret data or geometry using correct units.
- Show full calculations and verify answers.
Types of data
Data are observations or measurements. Categorical data describe groups; numerical data are quantities.
Worked example
Favourite subject is categorical; test score is numerical.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Frequency tables
Frequency counts how often a value or category appears.
Worked example
For scores 2,2,3,4,4,4, the frequency of 4 is 3.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Mean
Arithmetic mean is the sum of observations divided by their count.
Worked example
For 4,6,8, mean=(4+6+8)/3=6.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Median
Sort the data; take the middle value, or average the two middle values for even-sized datasets.
Worked example
For 2,3,7,9, median=(3+7)/2=5.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Mode
The mode is the most frequent value; data may have one, multiple or no unique mode.
Worked example
2,2,3,3,4 has modes 2 and 3.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Range
Range equals maximum minus minimum and measures overall spread.
Worked example
For 5,9,11, range=11−5=6.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Weighted mean
A weighted mean uses the sum of value×weight divided by total weight.
Worked example
Scores 70 (weight 2) and 80 (weight 3): (140+240)/5=76.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Combined mean
Combined mean=(n₁m₁+n₂m₂)/(n₁+n₂), where n values are group sizes.
Worked example
Two students average 60 and three average 80: combined mean 72.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Why simple averaging can fail
The unweighted average of two group means is correct only for equal group sizes.
Worked example
(60+80)/2=70 differs from the correctly weighted 72.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Changing data
Adding c to each observation increases the mean and median by c; multiplying by positive k scales them by k.
Worked example
Adding 5 to 4,6,8 changes mean from 6 to 11.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Outliers
Extreme values can affect the mean strongly while the median is often less sensitive.
Worked example
1,2,3,4,100 has mean 22 but median 3.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Bar charts
A bar chart displays frequencies of categories; a clear axis and scale prevent misleading comparisons.
Worked example
A category with frequency 12 should have a bar twice one with frequency 6 if the baseline is zero.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Histograms
Histograms represent grouped numerical data using adjacent intervals; with unequal class widths use frequency density.
Worked example
Intervals [0,10),[10,20) can be displayed as adjacent bars.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Interpreting graphs
Check axis scales, sample size, labels and missing categories before drawing conclusions.
Worked example
A truncated vertical axis can exaggerate a small difference.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Data quality
Missing values, duplicates, inconsistent units and biased samples can distort results.
Worked example
Mixing centimetres and metres without conversion gives incorrect averages.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Frequency-based mean
For values xᵢ with frequencies fᵢ, mean=Σ(fᵢxᵢ)/Σfᵢ.
Worked example
Values 1,2,3 with frequencies 2,1,1 have mean (2+2+3)/4=1.75.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Common errors
Always sort before finding median; do not average group means without weights.
Worked example
For 2,10,3, median is 3, not 10.
Reasoning check
Explain each step, state assumptions and verify the result independently where possible.
Key formulas and concepts
| Topic | Explanation | Worked illustration |
|---|---|---|
| Types of data | Data are observations or measurements. Categorical data describe groups; numerical data are quantities. | Favourite subject is categorical; test score is numerical. |
| Frequency tables | Frequency counts how often a value or category appears. | For scores 2,2,3,4,4,4, the frequency of 4 is 3. |
| Mean | Arithmetic mean is the sum of observations divided by their count. | For 4,6,8, mean=(4+6+8)/3=6. |
| Median | Sort the data; take the middle value, or average the two middle values for even-sized datasets. | For 2,3,7,9, median=(3+7)/2=5. |
| Mode | The mode is the most frequent value; data may have one, multiple or no unique mode. | 2,2,3,3,4 has modes 2 and 3. |
| Range | Range equals maximum minus minimum and measures overall spread. | For 5,9,11, range=11−5=6. |
| Weighted mean | A weighted mean uses the sum of value×weight divided by total weight. | Scores 70 (weight 2) and 80 (weight 3): (140+240)/5=76. |
| Combined mean | Combined mean=(n₁m₁+n₂m₂)/(n₁+n₂), where n values are group sizes. | Two students average 60 and three average 80: combined mean 72. |
| Why simple averaging can fail | The unweighted average of two group means is correct only for equal group sizes. | (60+80)/2=70 differs from the correctly weighted 72. |
| Changing data | Adding c to each observation increases the mean and median by c; multiplying by positive k scales them by k. | Adding 5 to 4,6,8 changes mean from 6 to 11. |
| Outliers | Extreme values can affect the mean strongly while the median is often less sensitive. | 1,2,3,4,100 has mean 22 but median 3. |
| Bar charts | A bar chart displays frequencies of categories; a clear axis and scale prevent misleading comparisons. | A category with frequency 12 should have a bar twice one with frequency 6 if the baseline is zero. |
| Histograms | Histograms represent grouped numerical data using adjacent intervals; with unequal class widths use frequency density. | Intervals [0,10),[10,20) can be displayed as adjacent bars. |
| Interpreting graphs | Check axis scales, sample size, labels and missing categories before drawing conclusions. | A truncated vertical axis can exaggerate a small difference. |
| Data quality | Missing values, duplicates, inconsistent units and biased samples can distort results. | Mixing centimetres and metres without conversion gives incorrect averages. |
| Frequency-based mean | For values xᵢ with frequencies fᵢ, mean=Σ(fᵢxᵢ)/Σfᵢ. | Values 1,2,3 with frequencies 2,1,1 have mean (2+2+3)/4=1.75. |
| Common errors | Always sort before finding median; do not average group means without weights. | For 2,10,3, median is 3, not 10. |
Practice exercise A — Definitions
- Define types of data, state any necessary conditions, and give a valid example.
- Define frequency tables, state any necessary conditions, and give a valid example.
- Define mean, state any necessary conditions, and give a valid example.
- Define median, state any necessary conditions, and give a valid example.
- Define mode, state any necessary conditions, and give a valid example.
- Define range, state any necessary conditions, and give a valid example.
- Define weighted mean, state any necessary conditions, and give a valid example.
- Define combined mean, state any necessary conditions, and give a valid example.
- Define why simple averaging can fail, state any necessary conditions, and give a valid example.
- Define changing data, state any necessary conditions, and give a valid example.
- Define outliers, state any necessary conditions, and give a valid example.
- Define bar charts, state any necessary conditions, and give a valid example.
Practice exercise B — Application
- Verify the worked example: Favourite subject is categorical; test score is numerical. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: For scores 2,2,3,4,4,4, the frequency of 4 is 3. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: For 4,6,8, mean=(4+6+8)/3=6. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: For 2,3,7,9, median=(3+7)/2=5. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: 2,2,3,3,4 has modes 2 and 3. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: For 5,9,11, range=11−5=6. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: Scores 70 (weight 2) and 80 (weight 3): (140+240)/5=76. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: Two students average 60 and three average 80: combined mean 72. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: (60+80)/2=70 differs from the correctly weighted 72. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: Adding 5 to 4,6,8 changes mean from 6 to 11. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: 1,2,3,4,100 has mean 22 but median 3. Change the values or assumptions and solve the new problem with all steps.
- Verify the worked example: A category with frequency 12 should have a bar twice one with frequency 6 if the baseline is zero. Change the values or assumptions and solve the new problem with all steps.
Practice exercise C — Reasoning
- Explain why a central formula or procedure is valid.
- Construct a counterexample to an incorrect generalisation.
- Compare two concepts and their use cases.
- Develop a real-world application with realistic units.
- Find and correct a plausible mistake.
- Check an answer by a second method.
- Construct and solve a three-step original problem.
- Explain the main concept to another student with a diagram or table.
Frequently asked questions and answers
What is types of data? Give an example.
Data are observations or measurements. Categorical data describe groups; numerical data are quantities. Example: Favourite subject is categorical; test score is numerical.
What is frequency tables? Give an example.
Frequency counts how often a value or category appears. Example: For scores 2,2,3,4,4,4, the frequency of 4 is 3.
What is mean? Give an example.
Arithmetic mean is the sum of observations divided by their count. Example: For 4,6,8, mean=(4+6+8)/3=6.
What is median? Give an example.
Sort the data; take the middle value, or average the two middle values for even-sized datasets. Example: For 2,3,7,9, median=(3+7)/2=5.
What is mode? Give an example.
The mode is the most frequent value; data may have one, multiple or no unique mode. Example: 2,2,3,3,4 has modes 2 and 3.
What is range? Give an example.
Range equals maximum minus minimum and measures overall spread. Example: For 5,9,11, range=11−5=6.
What is weighted mean? Give an example.
A weighted mean uses the sum of value×weight divided by total weight. Example: Scores 70 (weight 2) and 80 (weight 3): (140+240)/5=76.
What is combined mean? Give an example.
Combined mean=(n₁m₁+n₂m₂)/(n₁+n₂), where n values are group sizes. Example: Two students average 60 and three average 80: combined mean 72.
What is why simple averaging can fail? Give an example.
The unweighted average of two group means is correct only for equal group sizes. Example: (60+80)/2=70 differs from the correctly weighted 72.
What is changing data? Give an example.
Adding c to each observation increases the mean and median by c; multiplying by positive k scales them by k. Example: Adding 5 to 4,6,8 changes mean from 6 to 11.
What is outliers? Give an example.
Extreme values can affect the mean strongly while the median is often less sensitive. Example: 1,2,3,4,100 has mean 22 but median 3.
What is bar charts? Give an example.
A bar chart displays frequencies of categories; a clear axis and scale prevent misleading comparisons. Example: A category with frequency 12 should have a bar twice one with frequency 6 if the baseline is zero.
What is histograms? Give an example.
Histograms represent grouped numerical data using adjacent intervals; with unequal class widths use frequency density. Example: Intervals [0,10),[10,20) can be displayed as adjacent bars.
What is interpreting graphs? Give an example.
Check axis scales, sample size, labels and missing categories before drawing conclusions. Example: A truncated vertical axis can exaggerate a small difference.
What is data quality? Give an example.
Missing values, duplicates, inconsistent units and biased samples can distort results. Example: Mixing centimetres and metres without conversion gives incorrect averages.
Revision checklist
- I can explain and apply types of data.
- I can explain and apply frequency tables.
- I can explain and apply mean.
- I can explain and apply median.
- I can explain and apply mode.
- I can explain and apply range.
- I can explain and apply weighted mean.
- I can explain and apply combined mean.
- I can explain and apply why simple averaging can fail.
- I can explain and apply changing data.
- I can explain and apply outliers.
- I can explain and apply bar charts.
- I can explain and apply histograms.
- I can explain and apply interpreting graphs.
- I can explain and apply data quality.
- I can explain and apply frequency-based mean.
- I can explain and apply common errors.
Official source and accuracy note
Official NCERT textbook portal. This original guide is not a verified answer key to every numbered NCERT exercise.
Frequently asked questions
What is types of data? Give an example.
Data are observations or measurements. Categorical data describe groups; numerical data are quantities. Example: Favourite subject is categorical; test score is numerical.
What is frequency tables? Give an example.
Frequency counts how often a value or category appears. Example: For scores 2,2,3,4,4,4, the frequency of 4 is 3.
What is mean? Give an example.
Arithmetic mean is the sum of observations divided by their count. Example: For 4,6,8, mean=(4+6+8)/3=6.
What is median? Give an example.
Sort the data; take the middle value, or average the two middle values for even-sized datasets. Example: For 2,3,7,9, median=(3+7)/2=5.
What is mode? Give an example.
The mode is the most frequent value; data may have one, multiple or no unique mode. Example: 2,2,3,3,4 has modes 2 and 3.
What is range? Give an example.
Range equals maximum minus minimum and measures overall spread. Example: For 5,9,11, range=11−5=6.
What is weighted mean? Give an example.
A weighted mean uses the sum of value×weight divided by total weight. Example: Scores 70 (weight 2) and 80 (weight 3): (140+240)/5=76.
What is combined mean? Give an example.
Combined mean=(n₁m₁+n₂m₂)/(n₁+n₂), where n values are group sizes. Example: Two students average 60 and three average 80: combined mean 72.
What is why simple averaging can fail? Give an example.
The unweighted average of two group means is correct only for equal group sizes. Example: (60+80)/2=70 differs from the correctly weighted 72.
What is changing data? Give an example.
Adding c to each observation increases the mean and median by c; multiplying by positive k scales them by k. Example: Adding 5 to 4,6,8 changes mean from 6 to 11.
What is outliers? Give an example.
Extreme values can affect the mean strongly while the median is often less sensitive. Example: 1,2,3,4,100 has mean 22 but median 3.
What is bar charts? Give an example.
A bar chart displays frequencies of categories; a clear axis and scale prevent misleading comparisons. Example: A category with frequency 12 should have a bar twice one with frequency 6 if the baseline is zero.
What is histograms? Give an example.
Histograms represent grouped numerical data using adjacent intervals; with unequal class widths use frequency density. Example: Intervals [0,10),[10,20) can be displayed as adjacent bars.
What is interpreting graphs? Give an example.
Check axis scales, sample size, labels and missing categories before drawing conclusions. Example: A truncated vertical axis can exaggerate a small difference.
What is data quality? Give an example.
Missing values, duplicates, inconsistent units and biased samples can distort results. Example: Mixing centimetres and metres without conversion gives incorrect averages.
